> The key point is that it puts the onus on individual men, not on society at large.
A gentle nudge, I hope.
I'm not sure how the examples you give place any undue onus on individual men, nor how they absolve society at large.
I read both examples as gesturing at how the society socializes boys and what it expects of men. I doubt many would suggest individual men are making abstract, rational, uninfluenced, ruggedly-individual decisions to prefer suicide over seeking mental healthcare.
We're all responsible for how we expect people to perform their gender (and how our personal expectations factor in to what society at large expects), but the most-direct way to change those expectations is to reject them at scale.
In this case, men have the greatest leverage to initiate change. The point isn't that millions of men need to individually make a quiet decision to seek mental healthcare in secret--it's that millions of men can change the weather here if they individually choose to reject traditional expectations (including the reigning hyperagent ideology you cite) and make space for others to follow.
I hope it's obvious that retooling our gendered expectations of men (as a society) is a thread that's at least worth pulling on with respect to reducing male suicide rates. I hope it's also clear that a lot of individual men would have roles to play in that process.
You're right that it isn't the whole story. Creating the preconditions for men to readily seek mental healthcare doesn't matter much if the care isn't obtainable for other reasons. But I think it's a reasonable focus until there's good evidence that men are motivated to seek care that isn't ~accessible.
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Few social problems are best addressed with an answer from one perspective.
Some people over-flatten the gender pay gap, for example, into something inflicted on women by a culture that doesn't value them like it does men. Some people over-flatten it into something women inflict on themselves when they aren't as likely to negotiate a higher salary, ask for raises, job-hop, work in more dangerous industries, put in gobs of overtime, pursue management positions, get STEM degrees, and so on.
Some read the former as shirking personal responsibility and begging society for equal outcomes. Some people read the latter as putting the onus on individual women, not on society at large.
We can break out of frames like these and focus on helping people flourish. Some of us may not be able to walk and chew gum--but some of us can walk, and some of us can chew gum.